METU
Department of Mathematics
Math 748
Symplectic Topology
Fall 2014
S.Finashin
Test 1
(Take-home)
Q1. (5 pts) Let ω = e∗1 ∧ e∗2 + 2e∗2 ∧ e∗3 − (e∗1 + e∗3 ) ∧ e∗4 + 5e∗3 ∧ e∗1 ∈ Λ2 (R4 )∗ , where
{ei } is a basis of R4 .
(1) Find a matrix (ωij ) of ω, as it is viewed as a bilinear form in the given basis
of R4 .
(2) Find the images of vectors ei under the map ω
e.
(3) Determine if ω is non-degenerate.
(4) Find ω ∧ ω.
(5) Find the Pfaffian of the matrix (ωij ).
Q2. (3 pts) Given a symplectic form ω = e∗1 ∧ e∗2 + e∗2 ∧ e∗3 + e∗3 ∧ e∗4 in R4
(1) Find ω(v, w), where v = (1, 1, 1, 2) and w = (1, 3, 1, 3). Is span(v, w) a
Lagrangian plane ?
(2) Find a Lagrangian plane contained is the subspace span(e1 , e2 , e3 ) (that is,
defined by equation x4 = 0).
(3) Find a Lagrangian plane containing vector e1 + e2 + e3 + e4 .
Q3. (3 pts) Assume that b : V ×V → R is a nondegenerate bilinear (for simplicity,
symmetric or alternating) form in a vector space V of dimension n, and
W b = {v ∈ V | b(v, w) = 0, ∀w ∈ W }.
(1) Show that the restriction b|W is degenerate if and only if there exists v ∈ W ,
v ̸= 0, such that b(v, w) = 0 for all w ∈ W . And the latter is equivalent to
that W ∩ W b ̸= {0}.
(2) Assuming that b|W is non-degenerate, show that for any v ∈ V there exists
w ∈ W such that v − w ∈ W b .
(3) Show using the results above that V is a direct sum of W and W b if b|W is
non-degenerate, and deduce that dim W b + dim W = n.
Q4. (2 pts) Consider a subspace W ⊂ V in a vector space V and define
W ⊥ = {α ∈ V ∗ | α(v) = 0 ∀v ∈ W } ⊂ V ∗ .
Prove that W ×W ⊥ is a Lagrangian subspace of V ×V ∗ with respect to the canonical
symplectic form ωcan defined on V × V ∗ as follows:
ωcan ((v, α), (w, β)) = β(v) − α(w)
Q5. (4 pts) Prove that the intersection of any pair of the groups O(R2n ), GL(Cn ),
and Sp(R2n ) is contained in the third group. Deduce that the intersection of any
pair of these groups is the unitary group U(Cn ).
Recall the definitions: for a linear operator T : V → V
(1) T ∈ O(V ) (T is orthogonal with respect to an inner product (⟨∗, ∗⟩) in V ), if
⟨T v, T w⟩ = ⟨v, w⟩ ∀v, w ∈ V,
(2) T ∈ GL(V ) (T
is complex with respect to a complex structure
J : V → V ), if
T (Jv) = J(T v) ∀v ∈ V,
(3) T ∈ Sp(V ) (T
is symplectic with respect to a symplectic structure
ω ), if
ω(T v, T w) = ω(v, w) ∀v, w ∈ V.
Hint:
Note also that the complex structure in
R2n ∼
= Cn , the dot product and the canonical
symplectic form are related by the relation
ω(v, w) = ⟨v, Jw⟩, ∀v, w ∈ R2n